$($i$)$ Sandra is considering purchasing a financial asset that will pay $$3000$ per year for $8$ years, with the first payment $1$ year from now. The required rate of return is $12\%$ per year. How much should she pay for this asset?

$[$Hint: you are finding the PV for an ordinary annuity$]$

$4$

$($ii$)$ The People's Bank offers to pay you $9\%$ per annum compounded quarterly. If you invest $$40000,$ what is the value of the investment in $4$ years if interests are reinvested at $9\%?$ What would have been the value of the investment if the interest was compounded continuously?

$[4]$

$($iii$)$ For your retirement today, your company is offering you an immediate lump sum of $$0\mathrm{.}9$ million or an annual annuity of $20$ payments of $$150000$ starting today. Your bank is offering you an interest rate of $8\%$ per year compounded yearly. Which option has the greater present value? Justify your answer.

$($iv$)$ A pension fund anticipates that benefits of $$1500000$ per year must be paid to retirees. Retirements will commence $9$ years from now at $t=9.$ The benefit payments will continue until $t=25.$ What is the present value of the pension liability if the annual discount rate is $5\%$ compounded annually?

$[4]$

$($v$)$ The manager of a pension fund knows that the fund must make a payment of $8$ million dollars $12$ years from now. She wants to make an investment today so that it will grow to the required amount. The current interest rate is $8\%$ per year compounded every $2$ months. How much should she invest?

$[4]$